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Article

A Darboux theorem for derived schemes with shifted symplectic structure

Journal of the American Mathematical Society. 2019. No. 32. P. 399-443.
Brav C. I., Bussi V., Joyce D.

We prove a Darboux theorem for derived schemes with symplectic forms of degree k<0, in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme X with symplectic form omega' of degree k is locally equivalent to (Spec A, omega) for Spec(A) an affine derived scheme in which the cdga A has Darboux-like coordinates with respect to which the symplectic form omega is standard, and in which the differential in A is given by a Poisson bracket with a Hamiltonian function Phi of degree k+1.

When k=-1, this implies that a -1-shifted symplectic derived scheme (X,omega') is Zariski locally equivalent to the derived critical locus Crit(Phi) of a regular function Phi: U --> A^1 on a smooth scheme U. We use this to show that the classical scheme t_0(X) has the structure of an algebraic d-critical locus, in the sense of Joyce.

In a series of works, the authors and their collaborators extend these results to (derived) Artin stacks, and discuss a Lagrangian neighbourhood theorem for shifted symplectic derived schemes, and applications to categorified and motivic Donaldson-Thomas theory of Calabi-Yau 3-folds, and to defining new Donaldson-Thomas type invariants of Calabi-Yau 4-folds, and to defining Fukaya categories of Lagrangians in algebraic symplectic manifolds using perverse sheaves.